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Properties

Label 2-3e6-9.4-c1-0-9
Degree $2$
Conductor $729$
Sign $1$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.388 − 0.673i)2-s + (0.697 − 1.20i)4-s + (−1.18 + 2.05i)5-s + (1.25 + 2.16i)7-s − 2.64·8-s + 1.84·10-s + (−1.57 − 2.72i)11-s + (−0.668 + 1.15i)13-s + (0.972 − 1.68i)14-s + (−0.368 − 0.637i)16-s + 6.27·17-s + 8.06·19-s + (1.65 + 2.87i)20-s + (−1.22 + 2.11i)22-s + (−2.02 + 3.51i)23-s + ⋯
L(s)  = 1  + (−0.274 − 0.476i)2-s + (0.348 − 0.604i)4-s + (−0.531 + 0.920i)5-s + (0.472 + 0.818i)7-s − 0.933·8-s + 0.584·10-s + (−0.473 − 0.820i)11-s + (−0.185 + 0.321i)13-s + (0.259 − 0.450i)14-s + (−0.0920 − 0.159i)16-s + 1.52·17-s + 1.85·19-s + (0.370 + 0.642i)20-s + (−0.260 + 0.451i)22-s + (−0.422 + 0.732i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $1$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33535\)
\(L(\frac12)\) \(\approx\) \(1.33535\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.388 + 0.673i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.18 - 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.25 - 2.16i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.57 + 2.72i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.668 - 1.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.27T + 17T^{2} \)
19 \( 1 - 8.06T + 19T^{2} \)
23 \( 1 + (2.02 - 3.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.64 - 8.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.41 - 2.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.53T + 37T^{2} \)
41 \( 1 + (-3.55 + 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.16 + 2.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.30 - 3.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.135T + 53T^{2} \)
59 \( 1 + (-1.99 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.170 + 0.296i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.06 - 8.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.19T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (2.04 + 3.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.456 + 0.790i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.72T + 89T^{2} \)
97 \( 1 + (-2.99 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48497674859837223805207249839, −9.711369530476941186745967251336, −8.825569018312267029977458309378, −7.74660828152495639894381506681, −7.02778809174937824155214490060, −5.72302815693358956064591453170, −5.31764060693529762002078879341, −3.37238969029074383001250513887, −2.79088006352780943250900595982, −1.27366781146112692728748362775, 0.904576978184203258710874057489, 2.77988233808895844721951262178, 4.02646366980954968539104873945, 4.90463088935125427207128410992, 6.00301419852021267227880711350, 7.38609771380006485687486065108, 7.74554030972477484411307263097, 8.273862793842636761638912789584, 9.537640536794665421847928957509, 10.17215061917072289012461146074

Graph of the $Z$-function along the critical line